Finite Group Databasebeta

← #6437171

Subgroups — #6437171

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35343C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic, complemented
3#30235C212yesmaximal, complemented
4#30227C212yesmaximal, complemented
5#30217C212yesmaximal, complemented
6#30216C212yesmaximal, complemented
7#30196C212yesmaximal, complemented
8#30195C214yesmaximal, complemented
9#30194C212yesmaximal
10#23832C214yesmaximal, complemented
11#23829C212yesmaximal, complemented
12#23828C212yesmaximal, complemented
13#21406C212yesmaximal, complemented
14#21403C212yesmaximal, complemented
15#20367C211yesmaximal, characteristic, complemented
16C2×#529C2212yes—
17C2×C4×M16C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2868C2211yescharacteristic
20#2861C2212yes—
21#2859C2212yes—
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2858C2211yescharacteristic
25#2629C2212yes—
26#2628C2212yes—
27#2621C2211yescharacteristic
28#2620C2212yes—
29#2619C2211yescharacteristic
30#2602—22no—
31#2602—22no—
32#2601—22no—
33#2601—22no—
34#2570—22no—
35#2550—22no—
36#2441—22no—
37#2439—22no—
38#2331C2212yes—
39#2330C2212yes—
40#2328C2211yescharacteristic
41#2327C2211yescharacteristic
42#2326C2212yes—
43#2285C2212yes—
44C4×#117C2211yescharacteristic
45#2101C2214yes—
46#2101C2214yes—
47#2100C2214yes—
48#2099C2214yes—
49#2074C2214yes—
50#2073C2214yes—
51#2072C2214yes—
52#2071C2214yes—
53#2043C2214yes—
54#2042C2214yes—
55#2042C2214yes—
56#2041C2214yes—
57#2026C2214yes—
58#2025C2214yes—
59#2024C2214yes—
60#2023C2214yes—
61#1992C2212yes—
62#1992C2212yes—
63#1992C2212yes—
64#1992C2212yes—
65#1959C2212yes—
66#1959C2212yes—
67#1959C2212yes—
68#1959C2212yes—
69#1895C2214yes—
70#1894C2214yes—
71#1893C2214yes—
72#1892C2214yes—
73#1763C2214yes—
74#1762C2214yes—
75C4×#103C2214yes—
76C4×#102C2214yes—
77#1746C2212yes—
78#1746C2212yes—
79#1745C2212yes—
80#1745C2212yes—
81#1729C2212yes—
82#1729C2212yes—
83C22×M16C2312yes—
84C23×C8C2312yes—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89#529—28no—
90#529—28no—
91C2×#126—22no—
92C2×#125—22no—
93C2×#125C2312yes—
94C2×#125C2312yes—
95C2×#124C2312yes—
96C2×#124C2312yes—
97C2×#123—22no—
98C2×C22⋊Q8C2312yes—
99C2×#121C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×D4—22no—
102C2×C4×D4C2312yes—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C4×M16—24no—
117C2×#97—24no—
118C2×#97C2312yes—
119C2×#97C2312yes—
120C2×#97C2312yes—
121C2×C4×C8C2312yes—
122#400—22no—
123#399—22no—
124#398—24no—
125#398—22no—
126#397—22no—
127#396—22no—
128#396—22no—
129#396—24no—
130#395—22no—
131#394—22no—
132#394—22no—
133#394—22no—
134#393—22no—
135#392—22no—
136#391—24no—
137#391C2312yes—
138#391C2312yes—
139#390—22no—
140#390—22no—
141#390C2312yes—
142#390C2312yes—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—24no—
150#389C2314yes—
151#389C2314yes—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#387—24no—
159#387—24no—
160#387—22no—
161#387—22no—
162#386—22no—
163#386—22no—
164#385—22no—
165#385—22no—
166#384—22no—
167#384C2312yes—
168#384C2312yes—
169#384C2312yes—
170#384C2312yes—
171#382—22no—
172#382—22no—
173#382C2311yescharacteristic
174#382C2311yescharacteristic
175#382C2312yes—
176#380—22no—
177#380C2314yes—
178C4×C4⋊C4—22no—
179C4×C4⋊C4D412yes—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4C2314yes—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4D412yes—
184#376C2311yescharacteristic
185#376C2311yescharacteristic
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C22×Q8—22no—
201C22×Q8C2412yes—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C2412yes—
205C23×C4—22no—
206C23×C4C2×D412yes—
207C23×C4C2411yescharacteristic
208C2×M16—24no—
209C2×M16—42no—
210C2×M16—42no—
211C22×C8—24no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8C2414yes—
215C22×C8C2414yes—
216C4⋊Q8—44no—
217C4⋊Q8—42no—
218C4⋊Q8—84no—
219#127—42no—
220#126—48no—
221#125—24no—
222#125—24no—
223#125—42no—
224#125—42no—
225#125—84no—
226#124—24no—
227#124—24no—
228#124—44no—
229#123—48no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232#121—24no—
233#121—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—48no—
239#117—42no—
240#117—42no—
241#117—48no—
242#117—84no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2414yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D411yescharacteristic
280C2×C22⋊C4C2×D411yescharacteristic
281C2×C42—22no—
282C2×C42—24no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42#4114yes—
289C2×C42#4114yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#103—44no—
297#103—44no—
298#103—44no—
299#103—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303#102—44no—
304#97—24no—
305#97—24no—
306#97—24no—
307C4×C8—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95C2×D412yes—
321#95C2×D412yes—
322#95C2×D412yes—
323#95C2×D412yes—
324C24—22no—
325C24C22×D411yescharacteristic
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—44no—
331C2×Q8—84no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4—82no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4#12418yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4C2511yescharacteristic
367M16—48no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—48no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—84no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—44no—
426C42—84no—
427C42#12418yes—
428C42C2×#4112yes—
429C42C2×#4112yes—
430C42C2×#4112yes—
431C42C2×#4112yes—
432C23—22no—
433C23—22no—
434C23—21no—
435C23—21no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23C2×#12412yes—
444C23C23×D411yescharacteristic
445Q8—44no—
446Q8—44no—
447Q8—164no—
448D4—44no—
449D4—44no—
450D4—88no—
451D4—164no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—48no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—88no—
491C2×C4C2×#12414yes—
492C2×C4C2×#12412yes—
493C2×C4C2×#12414yes—
494C2×C4C22×#4111yescharacteristic
495C2×C4C22×#4111yescharacteristic
496C2×C4#58512yes—
497C2×C4#58512yes—
498C8—44no—
499C8—44no—
500C22—22no—
501C22—24no—
502C22—42no—
503C22—41no—
504C22—41no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22C22×#12411yescharacteristic
513C22#344911yescharacteristic
514C22#344911yescharacteristic
515C22#358512yes—
516C22#358612yes—
517C4—22no—
518C4—24no—
519C4—24no—
520C4—42no—
521C4—42no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—82no—
527C4—84no—
528C4—84no—
529C4—82no—
530C4—82no—
531C4#345512yes—
532C4#345512yes—
533C2—42no—
534C2—162no—
535C2#3463711yescharacteristic
536C2#6062112yes—
537C2#6101612yes—
538C2#6101911yescharacteristic
539C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #35294C2semidirect
3. #30235C2semidirect
4. #30227C2semidirect
5. #30217C2semidirect
6. #30216C2semidirect
7. #30196C2semidirect
8. #30195C2semidirect
10. #23832C2semidirect
11. #23829C2semidirect
12. #23828C2semidirect
13. #21406C2semidirect
14. #21403C2semidirect
15. #20367C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#353432
#352942
#302352
#302272
#302172
#302162
#301962
#301952
#301942
#238322
#238292
#238282
#214062
#214032
#203672